Prove that the height of a cylinder of maximum volume, inscribed in a sphere of radius 'a' is `2a//sqrt3`.

Explanation

To prove that the height of a cylinder of maximum volume inscribed in a sphere of radius 'a' is \( \frac{2a}{\sqrt{3}} \), we can follow these steps:

### Step 1: Define the Variables
Let the height of the cylinder be \( h \) and the radius of the cylinder be \( r \). The radius of the sphere is given as \( a \).

### Step 2: Relate the Cylinder and Sphere
The cylinder is inscribed in the sphere, which means that the diagonal of the cylinder forms a right triangle with the height and radius. According to the Pythagorean theorem, we can express this relationship as:
\[
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